Experiment 6
Bragg Scattering
Introduction
Within the Tel-X-Ometer, electrons are accelerated through a potential difference
of 30 kilovolts to an energy of 30 keV. The electrons are smashed into a copper target.
Electron transitions in the copper then produce x-rays with two characteristic
wavelengths. These x-rays, commonly called Kα and Kβ, have wavelengths of λα = 0.138
nm and λβ = 0.154 nm. The Kβ x-rays tend to be more intense than the Kα x-rays. In
addition, scattering and bremsstrahlung create a background of x-rays over a wide range
of wavelengths.
The x-ray beam produced is directed at a target crystal. The Tel-X-Ometer comes
with three targets: NaCl, KCl, and RbCl. The crystals can be rotated and the x-ray
intensity coming off the crystal measured over a wide range of angles. William Lawrence
Bragg and his father jointly received the Nobel Prize in 1915 for their discovery and use
of Bragg's Law.
Bragg's Law predicts constructive interference of x-rays reflecting off adjacent
layers of atoms in the crystals according to the formula
2 d sin θ = m λ
where d = separation distance between atom layers in the crystal
θ = incident and reflection angle of the x-rays, measured from the surface
of the crystal
m = order number = 1, 2, 3, . . .
λ = x-ray wavelength
1
Experiment
Record the observed counting rates, using the NaCl crystal as the target, as a
function of 2θ in the datasheet template. Notice that in the regions of 25-35o, 55-70o, and
90-115o we go in steps of 1o. Once the individual peaks in the counting rate are identified,
data will be taken in smaller steps, as small as 10 min or 15 min of arc. Record the data in
the space provided. Also make sure that you record the counting rate normalized to the
same counting period, e.g. counts/sec.
NOTE: If you are unable to take and record data, a fully filled out data sheet with data
taken between 2θ = 11° to 2θ = 124° will be provided to you.
If there is sufficient time available in the lab period, you may later replace the
NaCl target with KCl and RbCl crystals and repeat the measurements for the m=1 peak
and the more intense Kβ x-ray. Space is provided on the datasheet for these
measurements.
Study the data sheet and determine the angles – as accurately as you can – at
which maxima (constructive interference) occurred. You should find at least five clear
angles at which maximum intensities occur, the first two are around 2θ = 29° and 2θ =
31°.
Assuming the first two maxima you found correspond to m = 1 and the two
wavelengths λα and λβ given above, calculate d, the distance between the crystal layers in
NaCl. The crystal rotates only 1/2 as much as the control arm so you must divide the
listed 2θ angles by 2 to get the actual reflection angle θ.
The other angles at which you found maxima should correspond to higher-order
maxima. Calculate the wavelengths for these assuming m = 2 or m = 3. You should get
very nearly the same value for d as for m = 1; if not, try using a different m value or recheck your work.
2
(Q1)
How does your measured value of d compare with the accepted experimental
value? You may find such a value by searching for the lattice constant of various crystals.
Note, however, that the lattice constant is typically defined as 2*d.
(Q2)
Which x-ray wavelength was more intense (based on the number of counts per
second)?
(Q3)
Which wavelength and order seems to be missing in this data?
(Q4)
For the missing maximum of question 2, at which angle should the maximum
have occurred?
(Q5)
Examine the data, can you find evidence of the "missing maximum" of question 3
in the data?
(Q6)
Which measurement of d would you think is most accurate? The measurements
for m=1, or 2, or 3? The measurements for λα or λβ? Explain why.
Other Crystals
Using the more intense x-ray wavelength (see question 1), the m = 1 maximum
was measured for KCl and RbCl, the experimental results were
2θ = 28.4°
for KCl
2θ = 27.16°
for RbCl
or substitute your measured values.
Calculate the atomic spacing d for each of these crystals. Also look up the atomic
numbers of Cl, Na, K, and Rb using the periodic table.
(Q7)
Describe any pattern you discover regarding the crystal spacing and atomic
numbers for the three crystals.
These crystals all have a similar
structure, like cubes with alternating atoms at
each corner. Like that shown in the figure to
the right.
The Handbook of Physics (Springer 2002) gives values for the radii (distances to
the outermost electrons) of these atoms and their most common ions:
3
d
Cl
0.089 nm
Cl–
0.181 nm
Na
0.189 nm
Na+
0.097 nm
K
0.236 nm
K+
0.133 nm
Rb
0.248 nm
Rb+
0.147 nm
(Q8) Can you explain, in terms of atomic shells and electrical forces, why the atoms and
ions have different sizes?
(Q9)
If atoms behave like hard spheres with the radii listed above and if they are
packed together to the point of "touching" in the crystals, then what would you expect for
the layer spacing (d) of each crystal? Calculate for both the neutral atoms and ions.
(Q10) Which values, the d values for the neutral atoms or for the ions, better matches the
experimental results?
4
2q
15.00
20.00
25.00
26.00
27.00
28.00
29.00
30.00
31.00
32.00
33.00
34.00
35.00
40.00
45.00
50.00
55.00
56.00
57.00
58.00
59.00
60.00
61.00
62.00
63.00
64.00
65.00
66.00
67.00
68.00
69.00
70.00
75.00
80.00
85.00
90.00
Counts/s
2q
91.00
92.00
93.00
94.00
95.00
96.00
97.00
98.00
99.00
100.00
101.00
102.00
103.00
104.00
105.00
106.00
107.00
108.00
109.00
110.00
111.00
112.00
113.00
114.00
115.00
Counts/s
2q
Peak 3
Peak 4
Peak 1
Peak 5
Peak 2
Peak 6
5
Counts/s
Bragg Diffraction Data for:

25.00
26.00
27.00
28.00
29.00
30.00
31.00
32.00
33.00
34.00
35.00
Bragg Diffraction Data for:

25.00
26.00
27.00
28.00
29.00
30.00
31.00
32.00
33.00
34.00
35.00
Counts/s
Peak 2
Peak 2
6
Counts/s